In this paper, an SIRI epidemic model with age of infection and the proliferation of susceptible individuals with logistic growth is investigated. By using the theory of integral semigroup and Hopf bifurcation theory for semilinear equations with non-dense domain, it is shown that if the threshold parameter is greater than unity, sufficient condition is derived for the occurrence of the Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical results.
1.
Introduction
In the past decades, great attention has been paid by many researchers to SIR epidemic models [1,2,3,4,5,6,7], in which total host population is divided into three classes called susceptible (S), infective (I) and removed (R), and the immunity that is obtained upon recovery is assumed to be permanent. In [8], for herpes viral infections, considering the fact of recovered individuals may relapse with reactivation of latent infection and reenter the infective class, Tudor proposed the following SIRI epidemic model:
where S(t),I(t) and R(t) represent the number of susceptible individuals, infectious individuals and recovered individuals at time t, respectively. Assumptions made in the system (1.1) are homogeneous mixing and constant population size. The parameter A is the constant birth rate, μ is the natural death rate, β is the contact rate, and γ is the recovery rate from the infective class. It is assumed that an individual in the recovered class can revert to the infective class with a constant rate δ. Here δ>0 implies that the recovered individuals would lose the immunity, and δ=0 implies that the recovered individuals acquire permanent immunity.
We note that in system (1.1), the total population size is assumed to be constant. In reality, demographic features which allow the population size to vary should be incorporated into epidemiological models in some cases. For a relatively long-lasting disease or a disease with high death rate, the assumption of logistic growth input of the susceptible seems more reasonable [9]. In [10], Gao and Hethcote investigated a SIRS model with the standard incidence rate, and considered a demographic structure with density-dependent restricted population growth by the logistic equation. In [11], Li et al. studied a SIR epidemic model with logistic growth and saturated treatment, and the existence of the stable limit cycles was obtained. Rencently, there are growing interests in epidemiological models with demographic structures of logistic type [12,13,14,15,16].
It is worth noting that in the above models, the transmission coefficient is assumed to be constant, and the infected person has the same infectivity during their periodic infection. However, laboratory studies suggest that the infectivity of infectious individuals be different at the differential age of infection [17,18]. Further, it was reported in [19,20] that, age-structure of a population is an important factor which affects the dynamics of disease transmission. In [21], Magal et al. discussed an infection-age model of disease transmission, where both the infectiousness and the removal rate depend on the infection age. In [22], an age structured SIRS epidemic model with age of recovery is studied, and the existence of a local Hopf bifurcation is proved under certain conditions. In [23], Chen et al. considered an SIR epidemic model with infection age and saturated incidence, and established a threshold dynamics by applying the fluctuation lemma and Lyapunov functional.
Motivated by the works of Chen et al. [23], Gao and Hethcote[10], and Tudor [8], in the present paper, we are concerned with the effects of logistic growth and age of infection on the transmission dynamics of infectious diseases. To this end, we consider the following differential equation system:
with the boundary condition
and the initial condition
where X=R+×L1+(0,∞)×R+, L1+(0,∞) is the set of all integrable functions from (0,∞) into R+=[0,∞). In system (1.2), S(t) represents the number of susceptible individuals at time t, i(a,t) represents the density of infected individuals with infection age a at time t, and R(t) is the number of individuals who have been infected and temporarily recovered at time t. All parameters in system (1.2) are positive constants, and their definitions are listed in Table 1.
In the sequel, we further make the following assumption:
Assumption 1.1 β(a)∈L∞+((0,+∞),R), moreover
For convenience, we assume that
where e−(μ+γ)a is the probability of infected individual to survive to age a and τ>0, β∗>0.
The organization of this paper is as follows. In Section 2, we formulate system (1.2) as an abstract non-densely defined Cauchy problem. In Section 3, we study the existence of feasible equilibria of system (1.2). In Section 4, the linearized equation and the characteristic equation of system (1.2) at the interior equilibrium are investigated, respectively. In Section 5, by analyzing corresponding characteristic equation, we discuss the existence of Hopf bifurcation. In Section 6, numerical examples are carried out to illustrate the theoretical results, and sensitivity analysis on several important parameters is carried out.
2.
Transformation of the Cauchy problem
In this section, we formulate system (1.2) as an abstract non-densely defined Cauchy problem.
Firstly, we normalize τ in (1.2) by the timescaling and age-scaling
and consider the following distribution
Dropping the hat notation, system (1.2) becomes
with the boundary condition i(0,t)=τ[S(t)∫∞0β(a)i(a,t)da+δR(t)], and the initial condition S(0)=S0≥0,i(0,⋅)=i0(a)∈L1((0,+∞),R),R(0)=R0≥0, where the new function β(a) is defined by
and
here τ≥0,β∗>0.
Define
where
then the first and the third equations of system (2.1) can be rewritten as follows
where
and
here
Let
Accordingly, system (2.1) is equivalent to the following system:
where
and
We now consider the following Banach space
with the norm
Define the linear operator Lτ:D(Lτ)→X by
with D(Lτ)={0R3}×W1,1((0,+∞),R3)⊂X, and the operator F:¯D(Lτ)→X by
Therefore, the linear operator Lτ is non-densely defined due to
Letting x(t)=(0R3w(⋅,t)), system (2.3) is transformed into the following non-densely defined abstract Cauchy problem
Based on the results in [24] and [25], the global existence, uniqueness and positivity of solutions of system (2.4) are obtained.
3.
Existence of feasible equilibria
In this section, we study the existence of feasible equilibria of system (2.4).
Define the threshold parameter R0 by
Suppose that ˉx(a)=(0R3ˉw(a))∈X0 is a equilibrium of system (2.4). Then we have
which is equivalent to
By direct calculation, we obtain
where ˉS=∫+∞0u1(a,t)da,ˉR=∫+∞0u2(a,t)da. From (3.1), it is easy to show that
Integrating Eq (3.2), we get
and
We derive from the first and second equations of Eq (3.1) that
This, together with (3.4), yields
On substituting (3.6) into (3.3), we obtain that
It follows from (3.5) that
We therefore follows from (3.6) and (3.8) that
On substituting (3.7)–(3.9) into (3.2), we get
Based on the discussions above, we have the following result.
Lemma 3.1. System (2.4) always has the equilibrium
In addition, if R0>1, there exists a unique positive equilibrium
Correspondingly, for system (1.2), we have the following result.
Theorem 3.1. System (1.2) always has a disease-free steady state E0(K,0,0). If R0>1, system (1.2) has a unique endemic steady state E∗(S∗,i∗(a),R∗), where
4.
Characteristic equation
In this section, we investivate the linearized equation of (2.4) around the positive equilibrium ˉx∗(a), and the characteristic equation of (2.4) at ˉx∗(a), respectively.
Making the change of variable y(t):=x(t)−ˉx∗(a), system (2.4) becomes
Accordingly, the linearized equation of system (4.1) around the origin is
where
with
After that, system (4.1) can be rewritten as
where the linear operator L:=Lτ+τDF(ˉx∗(a)) and
satisfying F(0)=0,DF(0)=0.
In the following, we give the characteristic equation of (2.4) at the positive equilibrium. By means of the method used in [26], we obtain the following lemma.
Lemma 4.1. Let λ∈Ω={λ∈C:Re(λ)>−μτ},λ∈ρ(Lτ) and
with (αψ)∈X and (0R3φ)∈D(Lτ), where Lτ is a Hille-Yosida operator and
Let L0 be the part of Lτ in ¯D(Lτ), that is L0:=D(L0)⊂X→X. For (0R3φ)∈D(L0), we have
where ^L0(φ)=−φ′−τQφ with D(^L0)={φ∈W1,1((0,+∞),R3):φ(0)=0}.
Note that τDF(ˉx∗):D(Lτ)⊂X→X is a compact bounded linear operator. It follows from (4.4) that
Therefore
By using the perturbation results of [35], we get
Hence, we have the following result.
Lemma 4.2. The linear operator Lτ is a Hille-Yosida operator, and its part (Lτ)0 in ¯D(Lτ) satisfies
Set λ∈Ω. Since λI−Lτ is invertible, it follows that λI−Lτ is invertible if and only if I−τDF(ˉx∗)(λI−Lτ)−1 is invertible, and
We now consider
which yields
It is easy to show that
Taking the formula of DB(ˉw∗) into consideration, we obtain
where
and
From (4.5), whenever Δ(λ) is invertible, we have
Using a similar argument as in [27], it is easy to verify the following result.
Lemma 4.3. The following results hold
(i) σ(Lτ)∩Ω=σp(Lτ)∩Ω={λ∈Ω:det(Δ(λ))=0};
(ii) If λ∈ρ(Lτ)∩Ω, we have the formula for resolvent
where
with Δ(λ) and K(λ,ψ) given by (4.6) and (4.7).
Under Assumption 1.1, it therefore follows from (4.7) that
From (4.6), we obtain the characteristic equation of system (2.4) at the positive equilibrium ˉx∗(a) as follows:
where
here
Letting λ=τζ, then
It is easy to show that
5.
Existence of Hopf bifurcation
In this section, by applying Hopf bifurcation theory [26], we are concerned with the existence of Hopf bifurcation for the Cauchy problem (2.4) by regarding τ as the bifurcation parameter.
From (4.11), we have
For any τ≥0, if R0>1, it is easy to show that
Therefore, ζ=0 is not an eigenvalue of Eq (5.1). Furthermore, when τ=0, Eq (5.1) reduces to
A direct calculation shows that
and
Hence, by Routh-Hurwitz criterion, when τ=0, we see that the equilibrium ˉx∗(a) is locally asymptotically stable if R0>1; and ˉx∗(a) is unstable if R0<1.
Substituting ζ=iω(ω>0) into Eq (5.1) and separating the real and imaginary parts, one obtains that
Squaring and adding the two equations of (5.3), it follows that
where
Letting z=ω2, Eq (5.4) can be written as
Denote
and define
By a similar argument as in [28], we have the following result.
Lemma 5.1. [28]. For the polynomial equation (5.6), the following results hold true:
(i) If h0<0, then Eq (5.6) has at least one positive root.
(ii) If h0≥0 and Δ<0, then Eq (5.6) has no positive root.
(iii) If h0≥0 and Δ≥0, then Eq (5.6) has at least one positive root if one of z∗1>0 and h(z∗1)≤0.
Noting that
without loss of generality, we may assume that Eq (5.6) has two positive roots denoted respectively as z1 and z2. Then Eq (5.4) has two positive roots ωk=√zk(k=1,2). Further, from (5.3), we have
where k=1,2;j=0,1,⋯, and
Based on the above discussion, we have the following result.
Theorem 5.1. Let Assumption 1.1 and R0>1 hold. If ωk is a positive root of Eq (5.4) and q1≠0, then
Therefore ζ=iωk is a simple root of Eq (5.1).
Proof. It follows from (5.1) that
and
Suppose that dg(ζ)dζ|ζ=iωk=0, then
Separating the real and imaginary parts in Eq (5.8), we obtain
Squaring and adding the two equations of (5.9), we derive that
which mean that
Since ωk>0, it follows that
which leads to a contradiction. Hence, we have
Let ζ(τ)=α(τ)+iω(τ) be a root of Eq (5.1) satisfying α(τ(j)k)=0,ω(τ(j)k)=ωk, where
Lemma 5.2. Let Assumption 1.1 and R0>1 hold. If zk=ω2k,h′(zk)≠0 and q1≠0, then
and dReζ(τ)/dτ and h′(zk) have the same sign.
Proof. Differentiating the two sides of Eq (5.1) with respect to τ, we get
On substituting ζ=iωk into Eq (5.1), by calculating, we have
A direct calculation shows that
From Eq (5.4), we get
It therefore follows that
Since zk>0, we conclude that Re[dζ(τ)/dτ] and h′(zk) have the same sign.
Noting that when τ=0, the equilibrium x∗(a) of (2.4) is locally asymptotically stable if R0>1, from what has been discussed above, we have the following results.
Theorem 5.2. Let τ(j)k and ω0,τ0 be defined by (5.7) and (5.10), respectively. If Assumption 1.1 and R0>1, q1≠0 hold.
(i) the endemic steady state E∗ of system (1.2) is locally asymptotically stable for all τ≥0 if h0≥0 and Δ≤0.
(ii) the endemic steady state E∗ is asymptotically stable for τ∈[0,τ0) if h0<0 or h0≥0,Δ>0,z∗1>0 and h(z∗1)≤0.
(iii) system (1.2) undergoes a Hopf bifurcation at endemic steady state E∗ when τ=τ(j)k (j=0,1,2,⋯) if the conditions as stated in (ii) are satisfied and h′(zk)≠0.
6.
Numerical simulations
In this section, numerical simulations will be given to illustrate the theoretical results in Section 5. Further, sensitivity analysis is used to quantify the range of variability in threshold parameter and to identify the key factors giving rise to threshold parameter, which is helpful to design treatment strategies.
6.1. Dynamics of system (1.2)
In this section, we give a numerical example to illustrate the main results in Section 5.
Based on the research works of tuberculosis [29,30,32], parameter values of system (1.2) are summarized in Table 2, and the maximum infection age is 60. Denote the numbers of infected individuals at time t as I(t)=∫1000i(t,a)da, and
By calculation, we have R0=8.2379>1,h0=−1.1989×10−7<0 and h′(zk)=2.7846×10−4>0. In this case, system (1.2) has an endemic steady state E∗(0.6070,0.0879τe−0.3143τa,3.4534). By calculation, we obtain ω0=0.0565 and τ0=14.3556. By Theorem 5.1, we see that the endemic steady state E∗ is locally asymptotically stable if τ∈[0,τ0) and is unstable if τ>τ0. Further, system (1.2) undergoes a Hopf bifurcation at E∗ when τ=τ0. Numerical simulation illustrates the result above (see, Figures 1 and 2).
Remark. From Figure 1, we see that when the bifurcation parameter τ is less than the critical value τ0, the endemic steady state E∗ of system (1.2) is locally asymptotically stable. From Figure 2, we observe that E∗ losses its stability and Hopf bifurcation occurs when τ crosses τ0 to the right (τ>τ0). This implies that the age, i.e., infection period τ is the key factor that causes the endemic steady state E∗ to become unstable and the appearance of Hopf bifurcation.
6.2. Sensitivity analysis
Sensitivity analysis is used to quantify the range of variables in threshold parameter and identify the key factors giving rise to threshold parameter. In [33,34], Latin hypercube sampling (LHS) is found to be a more efficient statistical sampling technique which has been introduced to the field of disease modelling. LHS allows an un-biased estimate of the threshold parameter, with the advantage that it requires fewer samples than simple random sampling to achieve the same accuracy.
By analysis of the sample derived from Latin hypercube sampling, we can obtain large efficient data in respect to different parameters of R0. Figure 3 shows the scatter plots of R0 in respect to K, μ, δ and γ, which implies that δ is a positively correlative variable with R0, while μ is a negatively correlative variable. But the correlation between K, γ and R0 is not clear. In [34], Marino et al. mentioned that Partial Rank Correlation Coefficients (PRCCs) provide a measure of the strength of a linear association between the parameters and the threshold parameter. Furthermore, PRCCs are useful for identifying the most important parameters. The positive or negative of PRCCs respectively denote the positive or negative correlation with the threshold parameter, and the sizes of PRCCs measure the strength of the correlation. As can been seen in Figure 4, K, δ and γ are positively correlative variables with R0 while μ is negatively correlative variables.
By selecting different parameter values, we can explore the influence of the parameters μ and δ on the numbers of infected individuals at time t, which is denoted as I(t). As shown in Figure 5, increasing the natural death rate μ and decreasing the rate at which recovered individuals return to the infective class will have a positive impact on I(t) to some extent, which means that the influence of μ and δ on I(t) is consistent with that on R0.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 11871316, 11801340), the Natural Science Foundation of Shanxi Province (Nos. 201801D121006, 201801D221007).
Conflict of interest
The authors declare that they have no competing interests.